Compare the least squares estimators with the MLE.Problem 2-20ptsThe marketing department of Coca Colawanted to analyze the relationship between the price of Coke and the demand forit. They were concerned that if they continue increasing the price, people will switch to Pepsi. Calculate 𝛽0̂and 𝛽1̂for the following data and give interpretation for the results.

Problem 1 -20ptsConstruct the likelihood function 𝐿(𝛽0,𝛽1,𝜎2)of:𝑓(𝑌𝑖,𝛽0,𝛽1,𝜎2)=1√2𝜋𝜎2exp[−12(𝑌𝑖−𝛽0−𝛽1𝑥𝑖𝜎)2]Where 𝑌𝑖~𝑁(𝛽0+𝛽1𝑥𝑖,𝜎2)and estimate𝛽0,𝛽1and𝜎2in 𝑌𝑖=𝛽0+𝛽1𝑥𝑖+𝜀𝑖, where 𝜀𝑖~𝑁(0,𝜎2),using the MLE. Compare the least squares estimators with the MLE.Problem 2-20ptsThe marketing department of Coca Colawanted to analyze the relationship between the price of Coke and the demand forit. They were concerned that if they continue increasing the price, people will switch to Pepsi. Calculate 𝛽0̂and 𝛽1̂for the following data and give interpretation for the results.WeekPriceDemand in quantity117.7647225.3739322.8343417.7649525.3741Problem 3 (20 pts)

In problem 2 test the hypothesis that 𝛽1̂≠0at 5% level of significance, using the rejection region method. What conclusion you can provide, based on that test.Problem 4(20pts)For the data set in problem 2 construct95% confidence intervals for 𝛽1̂and explain the results. Does the interval include zero, if no explain what does this mean?Problem 5(20pts)Provide a proof that 𝛽1̂is an unbiased estimator for 𝛽1.Bonus question(10pts)Name all of the assumptions that you know for the linear regression model.